Topicm03463316f39b3822_1528449000663_0Topic

The cubic equations in the factored form

Levelm03463316f39b3822_1528449084556_0Level

Third

Core curriculumm03463316f39b3822_1528449076687_0Core curriculum

III. Equations and inequalities. The student:

6) solves equations in polynomial form of W(x)=0 for polynomials simplified to a factored form or the ones which can be simplified to a factored form by factoring out the common factor or using the grouping method.

Timingm03463316f39b3822_1528449068082_0Timing

45 minutes

General objectivem03463316f39b3822_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesm03463316f39b3822_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Writing cubic equations in the factored form.

3. Solving equations written in the factored form.

Learning outcomesm03463316f39b3822_1528450430307_0Learning outcomes

The student:

- writes cubic equations in the factored form,

- solves cubic equations written in the factored form.

Methodsm03463316f39b3822_1528449534267_0Methods

1. Diamond ranking.

2. Situation analysis.

Forms of workm03463316f39b3822_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm03463316f39b3822_1528450127855_0Introduction

The students work in groups using the diamond ranking technique to put the most important information about equations and techniques of solving them in order. In particular, they recollect the equivalent equations method and the ancients’ method of analysis (analisis antiquorum). Having finished, they present their posters. The teacher explains any doubts.

Procedurem03463316f39b3822_1528446435040_0Procedure

The teacher informs the students that the aim of the class will be solving cubic equations written in the factored form.

The students use the posters and on the analogy of other types of equations define the cubic equation.

The definition.

The cubic equation is the equation that can be transformed equivalently to the form $a{x}^3+b{x}^2+cx+d=0$, where $a\ne 0$.

The students work with their computers. They analyse the applet, paying special attention to the number of roots of the cubic equation. They formulate a conclusion.

Geogebra – graphs of the function $y=a{x}^3+b{x}^2+cx+d$ depending on the value of the coefficients.

[Geogebra applet]

The conclusion:
Every cubic equation with real coefficients has at least one real root.
Using the information, the students solve the task individually.m03463316f39b3822_1527752256679_0The conclusion:
Every cubic equation with real coefficients has at least one real root.
Using the information, the students solve the task individually.

Task

Solve the equations:

a) $x^3+125=0$,

b) $3x^3+192=0$.

Task

Solve the equations:

a) $\left(4-x\right)\left(x+5\right)\left(2x-3\right)=0$,

b) $2x^3-4x^2-5x+10=0$,

c) $x^3-4x=7x^2-28$.

Task

What are the relations between coefficients a, b, c, d of the equation $a{x}^3+b{x}^2+cx+d=0$, if numbers 1 and -1 are the roots of this equation?

Task

Check which of the numbers -3, -1, 1 makes the equation $x^3+8x^2+17x+6=0$.

Task

Find number a, knowing that number 3 is the root of equation $x^3+a{x}^2-5x+6=0$.

Discussion – what is the simplest method of solving the cubic equation?

The students decide that the simplest method is factorizing the equation to the maximum of the second degree. The students solve the tasks individually using the method.

Task

Give the largest of numbers being the roots of the equation:

a) $\left(x-4\right)\left(x+10\right)\left(x-2\right)=0$,

b) $x^3+49x=0$,

c) $x^3+2x^2-16x-32=0$.

Task

For which values of number $p$, $(p\in R)$ equation $(x-3)[x^2-2(2p+1)x+(p+2{)}^2]=0$ has two different solutions.

Extra task
The length of the edges of a cuboid, coming from one vertex, is expressed with natural numbers and equals (x+1), x, (x‑1). The volume of the cuboid equals 120. Find the sum of the lengths of the cuboid edges coming from one vertex.m03463316f39b3822_1527712094602_0Extra task
The length of the edges of a cuboid, coming from one vertex, is expressed with natural numbers and equals (x+1), x, (x‑1). The volume of the cuboid equals 120. Find the sum of the lengths of the cuboid edges coming from one vertex.

The students present the effects of their work, compare the results. The teacher explains the doubts and assesses the students’ work.

Lesson summarym03463316f39b3822_1528450119332_0Lesson summary

The students do the consolidation tasks. They cooperate to recapitulate the class and formulate the conclusions to be remembered.

The cubic equation is the equation that can be transformed equivalently to the form $a{x}^3+b{x}^2+cx+d=0$, where $a\ne 0$.

Every cubic equation with real coefficients has at least one real root.

Selected words and expressions used in the lesson plan

cubic equationcubic equationcubic equation

equationequationequation

factorizationfactorizationfactorization

graph of the cubic functiongraph of the cubic functiongraph of the cubic function

number of roots of the equationnumber of roots of the equationnumber of roots of the equation

roots of the equationroots of the equationroots of the equation